I'm trying to figure out for which algebraic structure
$$\underbrace{a+a+\cdots+a}_{n \text{-times}} = a * n$$
is true.
Now I know the question 'Is all multiplication repeated addition?' has been asked many times with the answer: NO because you cannot express non-integer (such as fractions or complex numbers) multiples as repeated addition. However I'm pretty sure that the reverse is true; that 'Repeated addition is always multiplication'
So my first thought was that Rings would be the appropriate algebraic structure for this, seeing that they have both addition and multiplication. However, the definition of a ring does not mention this property.
So I was thinking about this property and it seems like it holds for many rings, including the following:
- Integers
- Rationals
- Reals
- Complex numbers
- $m\times m$ Matrix Ring
- Polynomials where multiplication is scaling by a number
But... Then I ran into the Boolean ring where $\lor$ is the addition in the ring, and $\land$ is the multiplication. So...
$$???\,\,\underbrace{a\lor a\lor\cdots\lor a}_{n \text{-times}} = a \land n \,\,???$$
Now the problem is the type of the entity is totally different (true/false values). This doesn't even make sense; or does it? If this isn't true, then I'm not sure where that leaves me then, since it would imply that this property doesn't hold for rings in general. But then what does it hold for?
Any insight would be greatly apprecitated. :)